Question: $\dfrac{ 4i + 10j }{ 9 } = \dfrac{ 10i + k }{ 3 }$ Solve for $i$.
Solution: Multiply both sides by the left denominator. $\dfrac{ 4i + 10j }{ {9} } = \dfrac{ 10i + k }{ 3 }$ ${9} \cdot \dfrac{ 4i + 10j }{ {9} } = {9} \cdot \dfrac{ 10i + k }{ 3 }$ $4i + 10j = {9} \cdot \dfrac { 10i + k }{ 3 }$ Reduce the right side. $4i + 10j = {9} \cdot \dfrac{ 10i + k }{ {3} }$ $4i + 10j = {3} \cdot \left( 10i + k \right)$ Distribute the right side $4i + 10j = {3} \cdot \left( {10i} + {k} \right)$ $4i + 10j = {30}i + {3}k$ Combine $i$ terms on the left. ${4i} + 10j = {30i} + 3k$ $-{26i} + 10j = 3k$ Move the $j$ term to the right. $-26i + {10j} = 3k$ $-26i = 3k - {10j}$ Isolate $i$ by dividing both sides by its coefficient. $-{26}i = 3k - 10j$ $i = \dfrac{ 3k - 10j }{ -{26} }$ Swap signs so the denominator isn't negative. $i = \dfrac{ -{3}k + {10}j }{ {26} }$